Define the joint counting process
\(M_i (t) = N_i (t) - \int _0^t (h_i (s; \theta ) + \pi _i (s)) R_i (s) ds.\) 
where \(N_i (t)\)  is the counting process with jumps at all \(t \in {\bf T}_i \cup {\bf D}_i\) . Asymptotic consistency is guaranteed [1]} if
\(n^{-1} \tilde{U}_n (\theta _0 ) \overset{P}{\rightarrow } 0,\) 
and
\(n^{-1} \frac{d}{d \theta ^{\top }} \tilde{U}_n (\theta ) \big |_{\theta =\theta _0} = \Xi (\theta ),\) 
and
\(\lim _{n \rightarrow \infty } P \left( \left| n^{-1} \frac{d^2}{d \theta _j d\theta _k} \tilde{U}_n (\theta ) \right| < M \text{ for all }j,k \text{ and all } \theta \in \Theta _0 \right) = 1\) 
To prove (REF ), we decompose \(\tilde{U}_n (\theta _0)\)  into
three terms
\(n^{-1} \tilde{U}_n (\theta _0) = n^{-1} U_n (\theta _0 ) + n^{-1} \left(\hat{U}_n (\theta _0) - U_n (\theta ) \right) + n^{-1} \left(\tilde{U}_n(\theta _0) - \hat{U}_n (\theta _0) \right)\) 
where, recall, \(\hat{U}_n (\theta _0)\)  are the logistic score equations with \(\int _0^\infty X_i(t,s) \beta (s) ds\)  and \(\tilde{U}_n (\theta _0)\)  with the approximation \(C_{i,t}^\top J_{\phi , \psi } {\bf b}\) . [1]} (Theorem VI.1.1) show that \(n^{-1} U_n (\theta _0)\rightarrow 0\) . The second term
\(n^{-1} \left( \hat{U}_n (\theta _0) - U_n (\theta _0) \right) =\frac{1}{n} \sum _{i=1}^n \int _0^{\tau } \frac{h^{(1)} (s;\theta _0)}{\pi _i (s) + h (s;\theta _0)} d M_i (s).\) 

where \(h(\cdot ; \theta _0)\)  is given by (REF ). Lenglart's inequality implies the second term converges in probability to zero under Assumptions REF  and REF . The third term satisfies
\(&n^{-1} \left\Vert  \hat{U}_n (\theta _0) - \tilde{U}_n (\theta _0) \right\Vert  \\\le & M n^{-1} \sum _{i=1}^n \sum _{t \in {\bf T}_i \cup {\bf D}_i} \bigg \Vert  g\left( \tilde{W}_{i,t}^\top \theta + \sum _{k=K_x+1}^\infty \hat{c}_{i,k} (t) \int _{t-\Delta }^t \hat{\psi }_k (s) \phi (s)^\top b+ \log (\pi _i (t) ) \right) - g \left( \tilde{W}_{i,t}^\top \theta + \log (\pi _i (t)) \right) \bigg \Vert  \\&\le M n^{-1} \sum _{i=1}^n \sum _{t \in {\bf T}_i \cup {\bf D}_i}\sup _{x \in \mathbb {R}} \left\Vert  g^\prime \left( x \right)\right\Vert  \times \left\Vert  \sum _{k=K_x+1}^\infty \hat{c}_{i,k} (t)\sum _{l=1}^{K_b} \int _{t-\Delta }^t \hat{\psi }_k (s) \phi _l (s) b_l\right\Vert  \\&= \frac{M}{4} n^{-1} \sum _{i=1}^n \sum _{t \in {\bf T}_i \cup {\bf D}_i}\times \sum _{l=1}^{K_b} \left\Vert  \int _{t-\Delta }^t\left( X(t,s) - \sum _{k=1}^{K_x} \hat{c}_{i,k} (t) \psi _k (s) \right)\phi _l (s) b_l\right\Vert  \\&\rightarrow \frac{M}{4} \int _0^\tau \sum _{l=1}^{K_b} \left\Vert  \int _{t-\Delta }^t\left( X(t,s) - \sum _{k=1}^{K_x} \hat{c}_{i,k} (t) \psi _k (s) \right)\phi _l (s) b_l\right\Vert  (f_0(t) + f_1(t)) dt\) 

where \(M = \sup \Vert  \tilde{W}_{i,t} \Vert \) , \(g(x) = 1/(1+\exp (-x))\)  is the
expit function.
The second inequality is due to the Taylor remainder theorem. The third is due to \(g^\prime (x) = e^x/(1+e^x)^2 \le e^0/(1+e^0)^2 = 1/4\) , reordering the summation, and rewriting the error term in terms of the difference between \(X(t,s)\)  and the approximation with truncation level \(K_x\) . Letting \(n \rightarrow \infty \)  allows us to re-write the outside sums in terms of the integrated difference where the integral is with respect to the joint event and sampling distributions \((f_0(t) + f_1 (t))\) . Finally, under Assumptions REF , [3]} show
\(X(t,s) - \sum _{k=1}^{K_x} \hat{c}_{i,k} (t) \psi _k (s) \overset{P}{\rightarrow } 0\) 
as \(K_x \rightarrow \infty \) , which implies the third term goes to zero as the truncation level goes to infinity. The same argument can be used in conjunction under assumptions REF  and REF  to prove (REF ).
To prove (REF ), we start with equation (REF ), which we rewrite here for completeness:
\(\hat{U}_n (\theta ) = \sum _{i=1}^n \left[ \sum _{u \in {\bf T}_i} w_i(u; \theta )\frac{h^{(1)} (u; \theta )}{ h ( u; \theta )} - \sum _{u \in {\bf D}_i} w_i(u;\theta ) \frac{h_i^{(1)} (u; \theta )}{ \pi _i (u) } \right].\) 
Let \(N_i^{(e)} (t)\)  and \(N_i^{(s)} (t)\)  be the event and subsampling counting processes with jumps at \(t \in {\bf T}_i\)  and \({\bf D}_i\)  respectively. Then
\(\left| n^{-1} \frac{d^2}{d \theta _j d \theta _k} \tilde{U}_n (\theta ) \right| \le \frac{1}{n} \sum _{i=1}^n \int _0^\tau w_i (t;\theta ) \times H_{in} (t) R_i (t) dN_i^{(e)} (t) +\frac{1}{n} \sum _{i=1}^n \int _0^\tau \frac{w_i (t;\theta )}{\pi _i (t)} \times G_{in} (t) R_i (t) dN_i^{(e)} (t)\) 
where \(H_{in}(t)\)  and \(G_{in}(t)\)  are from Condition VI.1.1(E) in [1]}. The first term converges in probability to a finite quantity by [1]}. Since \(w_i(t;\theta )/\pi _i (t) = (\pi _i(t)+h_i(t;\theta ))^{-1}\) , the second term which simplifies to
\(\frac{1}{n} \sum _{i=1}^n \int _0^\tau \frac{G_{in} (t)}{\pi _i (t)+h_i(t;\theta )} R_i(t) dN_i^{(e)} (t)\le \frac{1}{L \cdot n} \sum _{i=1}^n \int _0^\tau G_{in} (t) R_i (t) dN_i^{(e)} (t)\) 
by Assumption REF  (i.e., the subsampling rate is lower bounded by \(L\) ). The right hand side is the optional variation of the local square integrable martingale
\(\frac{1}{L \cdot n} \sum _{i=1}^n \int _0^t G^{1/2}_{in} (s) dN_i^{(e)} (s)\) 
This martingale has predicable variation process
\(\frac{1}{L \cdot n} \sum _{i=1}^n \int _0^t G_{in} (s) \pi _i (s) R_i (s) ds\) 
Since both processes have the same limits, conditions VI.1.1(E) in [1]} and Assumption REF  imply that the second term on the right hand side of (REF ) also converges to a finite quantity as \(n \rightarrow \infty \)  for every \(K_x\)  which completes the proof.
This proves \(\hat{\theta }_n \overset{P}{\rightarrow } \theta _0\)  as \(n \rightarrow \infty \)  and \(K_x \rightarrow \infty \) . Note that the argument holds for any choice of weight functions \(w_i (t;\theta )\) ; implying convergence holds more broadly for all approximate score equations in (REF ).
